Schur multiplier

In mathematics, more specifically in group theory, the Schur multiplier (sometimes multiplicator), named after Issai Schur, is the second homology group of a group G with coefficients in the integers,

$H_{2}(G,{\mathbb {Z}})$.

If the group is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the generators, so that

$G\simeq F/R$,

then by Hopf's integral homology formula we have that the Schur multiplier is

$(R\cap [F,F])/[F,R]$,

where [A, B] is the group generated by the commutators

aba⁻¹b⁻¹

for a in A and b in B. It can also be expressed in terms of cohomology, as

$H^{2}(G,{\mathbb {C}}^{\times })$

where G acts trivially on the multiplicative group of the nonzero complex numbers.

Schur multipliers are of especial interest when G is a perfect group (a group equal to its own commutator subgroup). A perfect group has a unique perfect maximal central extension, or universal covering group, whose center contains the Schur multiplier and whose quotient by it is the perfect group.

The Schur multiplier is due to Issai Schur, and can be said to represent the beginning of group cohomology (or at least of the higher groups H² etc.).

References

• Karpilovsky, Gregory: The Schur Multiplier, Oxford University Press, 1987, ISBN 0-19-853554-6.